The topology and geometry of Bloch electrons
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摘要: 文章回顾了电子的拓扑几何理论发展的初期，大约二十多年的历史。首先介绍拓扑陈数在凝聚态物理中的两个重要应用。其一关于量子霍尔效应，绝缘条件下霍尔电导可以写成一个陈数拓扑不变量，从而解释实验结果的精确量子化。其二关于绝热泵浦，它描述布洛赫能带的绝热电流响应，与电子极化有密切联系。拓扑陈数是布里渊区上贝里曲率的积分，后者本身也有独立的物理意义。接着介绍贝里曲率对电子动力学的影响，包括反常速度和轨道磁化等概念。作者还将这个理论推广到多带情况，使其可以应用到自旋输运等现象。最后，文中展示了再量子化方法，从半经典模型来获得布洛赫电子的有效量子理论。在非相对论极限下，泡利—薛定谔方程可以看作是狄拉克电子在正能谱上的等效量子理论，其中的自旋轨道耦合即是一种几何物理效应。Abstract: We review the early development of electronic topological and geometric theory over a period of twenty some years, and explore two important applications of topological Chern numbers in condensed matter physics. The first is the quantum Hall effect, where the Hall conductivity can be written as a Chern number topological invariant under insulating conditions; its exact quantization found in experiment will be explained. The second is adiabatic pumping, which describes the adiabatic current response of Bloch bands and is closely related to electronic polarization. The topological Chern number is the integral of the Berry curvature over the Brillouin zone, wherein the latter has its own physical significance. We then describe the effect of Berry curvature on electron dynamics, including the anomalous velocity and orbital magnetization. We also generalize this theory to multi-band situations, which enables us to study spin transport phenomena. Finally, we demonstrate how to obtain an effective quantum theory by re-quantizing the semiclassical model. In the non-relativistic limit, the Pauli-Schrödinger equation can be seen as an equivalent quantum theory of Dirac electrons in the positive energy spectrum, where spin-orbit coupling is found to be a geometric effect.
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